ae554 applied orbital mechanics - metu | aerospace …ae554/sunum/ae554_orbit propa… · ·...
TRANSCRIPT
AE554 Applied Orbital Mechanics
Orbit Propagation
Egemen İmre
229/12/2007
Numerical Integration
� Solution of ODEs
� Many problems can be solved
� Generic methods exist
� Methods can be tailored for accuracy requirements (more later)
� Orbit Propagation: Integration of equations of motion
� Find position and velocity in time
“When your only tool is a hammer, every problem looks like a nail.”
Anonymous
329/12/2007
Propagation: Analytical vs. Numerical� Analytical
� Insight into the underlying dynamics
� Very fast
� Simplifications obligatory
� Complicated application of perturbing forces
� Very difficult (if not practically impossible) to include certain perturbations (3rd body etc)
� Numerical
� Fairly straightforward application of higher order geopotentials and other perturbativeforces
� Less insight into the dynamics
� Speed depends heavily on prediction timeframe and accuracy
� Long term stability depends primarily on integration technique
429/12/2007
Accelerations
Position and Velocity
integration
Anatomy of a Propagator
529/12/2007
Numerical Integration Basics
� The function f(t) is known (discrete values or continuous)but function Y may be too complicated to evaluate.
� Shown is first order approximation, all the second order and higher terms are neglected.
)(
)(
)(
2
0
1
0
tObtaY
tfY
tfy
k
i
ii
t
t
∆++∆≈
=
=
∑
∫
=
y
timet0
t1
∆t
629/12/2007
Numerical Integration: Concepts and Issues� In theory, as timestep goes to zero, integration error goes to zero as
well.
� In reality, there is a limit where further reduction in stepsize has the opposite effect!
� Error sources:
� Round-off error
� Truncation errors
� All are artefacts of “finite precision”
� Float vs. double vs. long double
� Function of stepsize (function evaluations)!
� Error
� Order in Taylor polynomial and appriximation comparable
729/12/2007
Numerical Integration: Concepts and Issues� Computational load
� Desktop computers and CPU farms
� Onboard applications?!
� 8bit &16bit vs 32bit & 64bit
� Force/function evaluations are expensive!
� Avoid them like plague!
� Principle of GIGO: Garbage in-garbage out
� Easy to set up a bad integration scheme
� Loss of precision (big number+small number)
� Verification?
� No truth model!
829/12/2007
Simple Numerical Integration Schemes (Explicit)
� Midpoint/rectangle rule
� Single force evaluation
� Zeroth order approx
� Assume function is constant within the step
� Trapezoid rule
� Double force evaluations
� First order approx
� Assume function changes linearly within the step
929/12/2007
Simple Numerical Integration Schemes (Explicit)
� Simpson’s rule
� Three force evaluations
� Second order approx
� Approximate the function with quadratic polynomials
� Composite rule
� n+1 force evaluations
Though we assumed equally spaced “substeps”, this is not necessarily the case!
1029/12/2007
Adaptive Schemes
� Previous schemes work well if the function is “well-behaved” and “smooth”.
� If this is not the case (such as a singularity or a “spike”) can be totally ignored by the numerical scheme.
� Solution: Adaptive scheme that “watches its step”.
� Estimate the rate of change of f(x)
� Check future values of f(x)
� Lower order scheme and compare
� Set a smaller timestep to capture the fast-changing behaviour
� Extra function evaluations and added complexity
� Existing schemes can be modified to work “adaptively”
1129/12/2007
The Story So Far...Applications to Orbit Propagation� Any of these methods can be used to integrate the
equations of motion
� Integrate acceleration/force >> velocity update
� Integrate velocity >> position update
� For higher the accuracy for a given duration:
� Smaller timesteps (more total force evaluations)
� Higher order integration (more total force evaluations)
� Find an optimal scheme
� Highly elliptic orbits: fast and slow dynamics at perigee and apogee
� Adaptive schemes ideal
1229/12/2007
Runge-Kutta Methods
� Runge-Kutta (RK) and adaptive RK schemes immensely popular!
� Single step method (no previous steps required)
� Ubiquitous 4th order RK scheme (RK4)
� Weighted average of four sample points per timestep
� 4th order scheme
� Error per step is O(h5)
� Total accumulated error is O(h4)
� Can be generalised to nth order
� Euler and midpoint are special cases of RK methods
1329/12/2007
Runge-Kutta Methods
� RK methods are versatile and easy to use, hence the popularity, particularly of higher order versions
� Not quite cheap!
� 4th order: 4 function evaluations per step
� 8th order: 13 function evaluations per step
� 10th order: 17 function evaluations per step
� RK8(7)-13 mnemonic (also a popular scheme)
� 8th order integration
� 7th order “embedded” scheme for “adaptivity”
� 13 force evaluations per step
1429/12/2007
Runge-Kutta-Nyström Method
� Rather than integrating acceleration twice to find the position update, integrate directly in a single step
� Solution of second order differential equations
� Particularly useful if the acceleration does not depend on the velocity (geopotential but NOT drag!)
� Save one function evaluation per timestep on RK6(4)!
� “Embedded” adaptive schemes available
1529/12/2007
Multistep Methods
� Adams-Bashforth and Adams-Bashforth-Moulton
1. Prediction: Estimate the solution at ti+1
2. Evaluation: Evaluate the fi+1 at ti+1
3. Correction: Apply correction to the solution at ti+1
4. Evaluation: Evaluate the fi+1 at ti+1 using the improved solution
� Attempt to minimise the number of function evaluations
� Iterative and (sort of) implicit method
� High-order explicit multistep methods (AB) more prone to “instability” than implicit methods (ABM)
� Computational errors dominate the result
1629/12/2007
More Multistep Methods
� Störmer-Cowell Methods
� Multistep methods for direct integration of second order differential equations (like RKN)
� Predictor-corrector scheme
� Gauss-Jackson or Second Sum Methods
� Modified Störmer-Cowell method (similar to ABM)
� Probably the best fixed-stepsize multistep method
� Very good stability properties in comparison to AB methods
1729/12/2007
Extrapolation Methods (Bulirsch-Stoer)� Single stepsize method
� Richardson extrapolation: Extrapolate to zero stepsize!
� Wacky idea – but it works!
� Use (a slightly modified) midpoint rule (with stepsize h)
� Evaluate it again with a different stepsize (h’)
� The two results using the two stepsizes can be extrapolated to another result that happens to have a stepsize of zero!
� Zero stepsize = zero error! (in theory, that is!)
� Use a polynomial extrapolation scheme
1829/12/2007
Extrapolation Methods (Bulirsch-Stoer)� Can be extended to any order via linear combinations of
results from different stepsizes
� When integrating over a stepsize of H, try stepsizes hi
)4for2(48... 32, 24, 16, 12, 8, 6, 4, 2, 2 ≥=== − innnn
Hh ii
i
i
� Similar to RK methods, a stepsize control can be devised by comparing the neighbouring estimates.
� When integrating over a stepsize of H, try stepsizes hi
1929/12/2007
Evaluation of the Methods
� Speed vs accuracy
� Slowed down by number of function evaluations
� If function evaluations are not that expensive, then arithmetic operaitons inside the scheme becomes important too
� Even with the adaptive methods, eccentric orbits are more expensive to calculate for a given accuracy
� Output points hamper the performance
� Use existing results as much as possible
� Montenbruck tests!
2029/12/2007
Conservation Principles
� The previous methods are good as general-purpose tools to solve ODEs
� But they have an utter disregard to the nature of the problem!
� Conservation laws regarding the problem – not quite conserved!
� No conservation, so what?!
� Results diverge from the truth!
� Similar to a dissipative system where no dissipation should occur!
� Semimajor axis decay and eccentricity change.
2129/12/2007
Conservation Principles
� Many problems in astrodynamics has conservation properties
� N-body problem
� Simulating the Solar System or a galaxy (stability?)
� Satellite orbit in a non-dissipative environment
� Non-LEO orbits
� For a satellite:
� Hamiltonian system (and Hamiltonian = Energy)
� Conservation of energy (as long as non-dissipative)
� Conservation of angular momentum (spherically symmetric force field i.e., no tesseral harmonics)
2229/12/2007
Hamiltonian: the concept
Hamiltonian = Energy (more or less!) – defines the motion
r = position, v = velocity (strictly speaking, momenta)
)(2
1)(),(),(
2rrrvrv UvRKH +=+=
� K = Keplerian (simple)
� R = higher order geopotentials (complicated!)
2329/12/2007
� Equations of Motion (Hamilton’s Equations)
� This definition conserves the energy. (symplecticness!)
� Separable to Keplerian and higher order geopotentials
Hamiltonian Dynamics
vr
r
rrv
v&& −=
∂
∂=
∂
∂==
∂
∂ )(UHH
2429/12/2007
Hamiltonian Dynamics
� Hamilton’s equations demonstrate a way to calculate the accelerations
� Conservation of energy
rr
vv
&&&
∂
∂+
∂
∂=
HHH = 0
vr
r
rrv
v&& −=
∂
∂=
∂
∂==
∂
∂ )(UHH
2529/12/2007
Hamiltonian Dynamics: Keplerian Case� Derive the equations for the Keplerian case (K is the
Keplerian Hamiltonian):
rvv
rrr
&&& ==∂
∂−==
∂
∂
−=
K
r
K
rvK
3
2
2
1
µ
µ
� Conservation of angular momentum:
03
=×+
−×=×+×=
×=
rrrrvrvrh
vrh
&&&&&
r
µ
2629/12/2007
Conserved Quantities: Reality
� For a more complicated potential (or, equivalently, force model) than Keplerian, conservation principles differ slightly.
� It can be shown that energy is constant for an axisymmetricgeopotential (i.e., zonal harmonics only). z component of the angular momentum is also constant.
� Potential is not time varying (latitudinal bands)
� For tesseral harmonics, energy variation is periodic
� Rotating longitudinal bands
� Third body effects induce periodic variations in force (or potential)
� Drag is a dissipative force and no conservation is possible
� But quite small!!
2729/12/2007
Symplecticness
� Symplecticness - Areas in phase space are conserved i.e., the conservation of the constants of motion (in practice!).
� Left: Forward Euler (Should be rotation only but distortion creeps in!)
� Right: Implicit Midpoint (Symplectic – Phase space area conserved!)
(Iserles and Zanna 1996)
p p
q q
2829/12/2007
Symplectic Integration
� Just another integration scheme – but with a twist!
� Unlike classical integrators, conserved quantites are conserved
� What we actually want is time-reversibility!
� Motion between (r,v)t0 and (r,v)t1 is a canonical transformation.
� Symplectic integrators better than classical integrators:
� An order of magnitude faster
� More reliable in long term
� No secular energy drift – can have fairly large local errors but better long term characteristics
2929/12/2007
Symplectic Integration
� Hamiltonian can usually be split into two parts (H=H0+H1)
� Kinetic + potential energy
� Kepler + J2
� Each part corresponds to a force (i.e., motion) via Hamilton’s equations
� We consider them in isolation
� The approximated motion corresponds to a Hamiltonian which is slightly different than the real Hamiltonian
� The error is a “surrogate Hamiltonian”
� The error is a “conserved quantity”!
� No “secular” errors – for a quasi-periodic system
3029/12/2007
Symplectic Integration: Leapfrog
� 2nd Order symplectic integration: Leapfrog
� For this example, applied to a Keplerian case
� Generalised leapfrog has an error of O(h3) in one step and O(h2) when repeated over a duration.
H1
12/1
2/1
1
2/1
2
1
)(
2
1
2
1
++
++
+
+=
∂
∂−=
+=
iii
iii
iii
h
Kh
h
vrr
r
rvv
vrr
Motion due to H0 (PE)
Motion due to H1 (KE)
3129/12/2007
Symplectic Integration: Higher Order Schemes� If H1 is smaller than H0, i.e. H1=O(ε) then the global error
becomes O(εh2)!
� Possible to derive higher order schemes
� Possible to derive adaptive schemes
� Stepsize must be kept constant!
� Time becomes another element of the position vector –with corresponding momentum term!
H1
H0
3229/12/2007
Conservation of Energy and Propagated Orbit
3329/12/2007
Energy: Hamiltonian
Accelerations: Hamilton’s Equations
Position and Velocity
Symplectic integration
Anatomy of a Symplectic Propagator
3429/12/2007
Symplectic Propagator: SPSAT
� SPSAT: Symplectic Orbit Propagator
� Integration scheme!
� Force model:
� Exact Keplerian dynamics
� “36x36” geopotential model
� Harris-Priester drag model (diurnal variations and stuff)
� Luni-solar attraction
� A bit dated WGS84 and very dated GEM10B geopotential models
� Simple conversion between rotating frame and inertial frame (fortesseral harmonics)
� Validation and verification is nigh on impossible!
� Set progressively smaller stepsizes
� Conservation of energy – fragile!
3529/12/2007
SPSAT Results – Orbit Stability
� Absolute position accuracy with respect to integration stepsize – near-circular case (36x36)
� Truth: 2000steps/orbit
3629/12/2007
SPSAT Results – Eccentricity
� Absolute position accuracy (200 steps/orbit) – near-circular and elliptic (e=0.35?) cases (36x36)
3729/12/2007
SPSAT Results – Absolute Energy
� Normalised absolute energy (200 steps/orbit) – near-circular and elliptic cases (36x36)
� Tesserals cause oscillation
3829/12/2007
SPSAT Results – Geopotentials
� Absolute position accuracy (100 steps/orbit) – near-circular case
� Truth: 1000 steps/orbit
3929/12/2007
4029/12/2007
� http://www.physics.drexel.edu/courses/CompPhys/Integrators/leapfrog/
� http://lec.ugr.es/~julyan/papers/rkpaper/node9.html